- Organizing data
- Developing theoretical principles
- Exploring hypotheticals
- Guiding policy
- Explaining observed patterns
In seeking to understand biological phenomena, models are tools for looking at the data.
They focus attention on the discrepancies between our hypotheses and reality.
How to quantify these discrepancies?
The key principle of statistics is that we must model the error.
The key principle is that we must model the error.
\[ \begin{aligned} N_{\emptyset{X}}(t)&=\text{cumulative number of births into X by time $t$}\\ N_{XY}(t)&=\text{cumulative number of X $\to$ Y movements by time $t$}\\ N_{Y\emptyset}(t)&=\text{cumulative number of Y $\to$ $\emptyset$ movements by time $t$}\\ \end{aligned} \]
\[ \begin{aligned} dN_{\emptyset{X}}&=B(t)\,dt \phantom{+\sigma_1(t)\,dW_1(t)}\\ dN_{XY}&=\lambda(Y)\,X\,dt \phantom{+\sigma_2(X,Y)\,dW_2(t)}\\ dN_{Y\emptyset}&=\mu\,Y\,dt \phantom{+\sigma_3(Y)\,dW_3(t)}\\ \end{aligned} \]
\[ \begin{aligned} dN_{\emptyset{X}}&=B(t)\,dt +\sigma_1(t)\,dW_1(t)\\ dN_{XY}&=\lambda(Y)\,X\,dt +\sigma_2(X,Y)\,dW_2(t)\\ dN_{Y\emptyset}&=\mu\,Y\,dt +\sigma_3(Y)\,dW_3(t)\\ \end{aligned} \]
\[dW_i\;\stackrel{i.i.d}{\sim}\;\mathrm{Normal}(0,\sqrt{dt})\]
\[ \begin{aligned} dX&=B(t)\,dt-dN_{XY}\\ &=B(t)\,dt-\lambda(Y)\,X\,dt+\sigma_1(t)\,dW_1(t)-\sigma_2(X,Y)\,dW_2(t)\\ dY&=dN_{XY}-dN_{Y\emptyset}\\ &=\lambda(Y)\,X\,dt+\sigma_2(X,Y)\,dW_2(t)+\sigma_3(X)\,dW_3(t)\\ \end{aligned} \]
What are the roles of seasonal and decadal climate drivers in the epidemiology of cholera?
What is the best vaccination strategy?
A large family of models:
These questions involve inescapable technical complications:
(Lavine, King, and Bjørnstad 2011)
(Domenech de Cellès, Magpantay, King, and Rohani 2016)
Why is pertussis resurgent?
Hypothetical answers include:
\[\lambda(I_1,I_2,t) = \frac{\beta(t)\,(I_1+\theta\,I_2)+\bar{\beta}\,\iota}{N}\]
Post-vaccination infections are observed at a reduced rate, \(\eta\).
(Magpantay, Domenech de Cellès, Rohani, and King 2016)
(Magpantay et al. 2016)
(Domenech de Cellès, Magpantay, King, and Rohani 2018)
(Domenech de Cellès et al. 2018)
(Domenech de Cellès et al. 2018)
(King, Nguyen, and Ionides 2016)
(King et al. 2016)
(King et al. 2016)
(King et al. 2016)
(King et al. 2016)
Definition: an algorithm has the plug-and-play property if it has no need to compute the latent process transition density.
Plug-and-play methods access the latent process model only via simulation.
This puts essentially no restrictions on the form of the models that can be entertained.
They are also called “simulation-based” methods.
(He, Ionides, and King 2010; King et al. 2016)
Variants of all of these are available in the pomp software package (King et al. 2016; [https://kingaa.github.io/pomp/]) and elswhere.
\[\lambda(t) = \frac{\beta(t)\,(I_1+\theta\,I_2)+\bar{\beta}\,\iota}{N}\]
Post-vaccination infections are observed at a reduced rate, \(\eta\).
(Magpantay et al. 2016)
Interpretation: in vaccinated hosts, infections are mild to asymptomatic, yet equally infectious
Flat profiles indicate lack of information in the data relative to the question.
In effect, the data refuse to answer the question.
(Magpantay et al. 2016)
(King, Ionides, Pascual, and Bouma 2008)
(King et al. 2008)
(King et al. 2008)
Is it necessary that all infected individuals be equally infectious?
(King et al. 2008)
among symptomatic infections, case fatality: \(0.34\pm 0.2\)
duration of immunity \(1.5 \pm 0.7~\text{yr}\)
(King et al. 2008)
(King et al. 2016)
The full joint density is:
\[f_{X,Y}(x,y;\theta) = f_0(x_0;\theta)\,\prod_{n=1}^N\!f_{n}(x_n|x_{n-1};\theta)\,g_{n}(y_n|x_n;\theta).\]
The likelihood function is the marginal density for \(Y\), evaluated at the data:
\[ \begin{split} \mathcal{L}(\theta)&=f_{Y}(y^*_1,\dots,y^*_n;\theta)\\ &=\int f_{X,Y}(x_0,\dots,x_N,y^*_1,\dots,y^*N;\theta)\, dx_1\cdots dx_n. \end{split} \]
Given parameters \(\theta\in\Theta\) and data \(y^*\in\mathcal{Y}\)
(Fasiolo, Pya, and Wood 2014; Wood 2010)
(Arulampalam, Maskell, Gordon, and Clapp 2002)
Because the particle filter gives as unbiased, though noisy, estimate of the likelihood, this MCMC converges to the correct target distribution.
(Andrieu, Doucet, and Holenstein 2010)
(Ionides, Bretó, and King 2006; Ionides, Nguyen, Atchadé, Stoev, and King 2015; King et al. 2016)
(Ionides et al. 2015)
Andrieu C, Doucet A, Holenstein R (2010). “Particle Markov Chain Monte Carlo Methods.” Journal of the Royal Statistical Society, Series B, 72(3), 269–342. https://doi.org/10.1111/j.1467-9868.2009.00736.x.
Arulampalam MS, Maskell S, Gordon N, Clapp T (2002). “A Tutorial on Particle Filters for Online Nonlinear, Non-Gaussian Bayesian Tracking.” IEEE Trans Signal Process, 50, 174–188. https://doi.org/10.1109/78.978374.
Domenech de Cellès M, Magpantay FMG, King AA, Rohani P (2016). “The Pertussis Enigma: Reconciling Epidemiology, Immunology, and Evolution.” Proceedings of the Royal Society of London. Series B,
Domenech de Cellès M, Magpantay FMG, King AA, Rohani P (2018). “The Impact of Past Vaccination Coverage and Immunity on Pertussis Resurgence.” Sci Transl Med, 10(434), eaaj1748. https://doi.org/10.1126/scitranslmed.aaj1748.
Fasiolo M, Pya N, Wood S (2014). “Statistical Inference for Highly Non-Linear Dynamical Models in Ecology and Epidemiology.” http://arxiv.org/abs/1411.4564
He D, Ionides EL, King AA (2010). “Plug-and-Play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study.” J R Soc Interface, 7, 271–283. https://doi.org/10.1098/rsif.2009.0151.
Ionides EL, Bretó C, King AA (2006). “Inference for Nonlinear Dynamical Systems.” Proc Natl Acad Sci, 103(49), 18438–18443. https://doi.org/10.1073/pnas.0603181103.
Ionides EL, Nguyen D, Atchadé Y, Stoev S, King AA (2015). “Inference for Dynamic and Latent Variable Models via Iterated, Perturbed Bayes Maps.” Proc Natl Acad Sci, 112(3), 719–724. https://doi.org/10.1073/pnas.1410597112.
King AA, Ionides EL, Pascual M, Bouma MJ (2008). “Inapparent Infections and Cholera Dynamics.” Nature, 454(7206), 877–880. https://doi.org/10.1038/nature07084.
King AA, Nguyen D, Ionides EL (2016). “Statistical Inference for Partially Observed Markov Processes via the R Package Pomp.” J Stat Softw, 69(12), 1–43. https://doi.org/10.18637/jss.v069.i12.
Lavine JS, King AA, Bjørnstad ON (2011). “Natural Immune Boosting in Pertussis Dynamics and the Potential for Long-Term Vaccine Failure.” Proceedings of the National Academy of Sciences of the U.S.A., 108(17), 7259–7264. https://doi.org/10.1073/pnas.1014394108.
Magpantay FMG, Domenech de Cellès M, Rohani P, King AA (2016). “Pertussis Immunity and Epidemiology: Mode and Duration of Vaccine-Induced Immunity.” Parasitology, 143, 835–849. https://doi.org/10.1017/S0031182015000979.
Wood SN (2010). “Statistical Inference for Noisy Nonlinear Ecological Dynamic Systems.” Nature, 466, 1102–1104. https://doi.org/10.1038/nature09319.